On a recursive construction of Dirichlet form on the Sierpinski gasket

Let Gamma(n) denote the n-th level Sierpiriski graph of the Sierpiriski gasket K. We consider, for any given conductance (a(0), b(0), c(0)) on Gamma(0), the Dirichlet form epsilon on K obtained from a recursive construction of compatible sequence of conductances (a(n), b(n), c(n)) or Gamma(n), n >= 0. We prove that there is a dichotomy situation: either a(0) = b(0) = c(0) and epsilon is the standard Dirichlet form, or a(0) > b(0) = c(0) (or the two symmetric alternatives), and epsilon is a non-self-similar Dirichlet form independent of a(0), b(0). The second situation has been studied in [13,9] as a one-dimensional asymptotic diffusion. The analytic approach here is more direct and yields sharper results; in particular, for the spectral property, we give a precise estimate of the eigenvalue distribution of the associated Laplacian, which improves a similar result in [9]. (C) 2019 Elsevier Inc. All rights reserved.